Chaining together routers?

I'm looking to get 12 or 13 machines all on one LAN. Unfortunately, it appears that 12+ port networking devices cost $$$$. Would it work if I were to take a 4 port router and plug another 4 port router into one of the ports, then another 4 port router into that one, and then another, to give me a total of 13 ports (considering the ports taken up by connecting the four routers)? I don't think routers work like that... please tell me I'm wrong. What should I do to overcome my networking dilemma?